Problem: $f(x,y) = -y + xy$ What is $\dfrac{\partial f}{\partial y}$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $x + y - 1$ (Choice B) B $0$ (Choice C) C $-1 + x$ (Choice D) D $y$
Solution: We want to find $\dfrac{\partial f}{\partial y}$, which is the partial derivative of $f$ with respect to $y$. When we take a partial derivative with respect to $y$, we treat $x$ as if it were a constant. Let's break $f(x, y)$ down term by term. $\begin{aligned} &\dfrac{\partial}{\partial y} \left[ -y \right] = -1 \\ \\ &\dfrac{\partial}{\partial y} \left[ xy \right] = x \end{aligned}$ Adding the terms back together, we get the partial derivative. In conclusion: $\dfrac{\partial f}{\partial y} = -1 + x$